
By Lemma 4.3 of [1]}, we know \(u^*\in L^2_{\mathcal {F}}(0,t;\mathbb {R}^m)\)  for any \(t>0\) . So by standard theory of SDE, we have
that \({\mathbb {E}}\big (\sup _{s\le t} X^*(s)^{2}\big )<\infty \)  for any \(t>0\) .
Because \(X^*(t\wedge \tau _n)^{2} \le \sup _{s\le t} X^*(s)^{2}\) , letting \(n\rightarrow \infty \)  in the above inequality and using the dominated convergence theorem and the monotone convergence theorem, we obtain
\(\beta {\mathbb {E}}\int _0^{t} (X^*)^2ds+{\mathbb {E}}\big (X^*(t)^2\big ) \ge x^2+\frac{\delta }{2}{\mathbb {E}}\int _0^{t}|u^*|^2ds.\) 
